CMI-IMSc Number Theory Seminar
Date: Tuesday, 25 February 2025
Time: 3:30 to 4:30 PM
Venue: Seminar Hall, CMI
Modular Galois representations with large Selmer $p$-rank
Anwesh Ray
Chennai Mathematical Institute.
25-02-25

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Abstract

The study of Galois representations arising from elliptic curves and modular forms is a central theme in modern number theory. In particular, the deformation theory of Galois representations has been instrumental in modularity results, most notably in Wiles’ proof of Fermat’s Last Theorem. Serre conjectured that every two-dimensional, odd, absolutely irreducible mod-p Galois representation arises as the reduction of a modular representation. In this work, we consider a prime p>3 and the case where the mod-p representation is reducible. Given any natural number N, we combine techniques from Iwasawa theory and Galois deformation theory to construct modular lifts whose associated Bloch-Kato Selmer group has p-rank at least N. Previous computational results have established this phenomenon for elliptic curves over Q when p≤13; we extend it to all primes p>3. This talk is based on joint work with Eknath Ghate.